Landau Lifshitz Equations (Frontiers of Research With the Chinese Academy of Sciences)

414 Pages·2008·5.27 MB·English

by  Boling Guo

LANDAU–LIFSHITZ
EQUATIONS
Frontiers of Research with the Chinese Academy of Sciences
Vol. 1 Landau-Lifshitz Equations
by Boling Guo and Shijin Ding
Frontiers of Research with the Chinese Academy of Sciences – Vol. 1
LANDAU–LIFSHITZ
EQUATIONS
Boling Guo
Beijing Institute of Applied Physics and

Computational Mathematics, China
Shijin Ding
South China Normal University, China
World Scientific
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Library of Congress Cataloging-in-Publication Data
Guo, Boling.
Landau-Lifshitz equations / by Boling Guo & Shijin Ding.
p. cm. -- (Frontiers of Research with the Chinese Academy of Sciences ; v. 1)
Includes bibliographical references.
ISBN-13: 978-981-277-875-8 (hardcover : alk. paper)
ISBN-10: 981-277-875-6 (hardcover : alk. paper)
1. Differential equations, Partial--Numerical solutions. 2. Maxwell equations--Numerical
solutions. 3. Geometry. 4. Mathematical physics. I. Ding, Shijin, 1959– II. Title.
QC776.G86 2008
518'.64--dc22
2007044801
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Preface
In studying the dispersive theory of magnetization of ferromagnets in 1935, Landau{
Lifshitz [102] proposed the equations of ferromagnetic spin chain which are important
magnetization equations, called Landau{Lifshitz equations now. Later on, such equa-
tions were also found in the condensed matter physics. In the 1960s, Soviet physicists
A. Z. Akhiezer, V. G. Beryahltar, S. V. Peletninskii studied spin wave, the equa-
tions of ferromagnetic chain and the traveling wave solutions in detail in their book
\Spin Waves"[3]. In 1974, K. Nakamura, T. Sasada [122] (cid:12)rst observed that there is
a soliton solution to the one-dimensional Landau{Lifshitz equations without Gilbert
damping. Then, many mathematicians and physicists studied the soliton theory of
Landau{Lifshitz equations using the approaches including inverse scattering method,
in(cid:12)nite many conservation laws, geometry expression method and gauge equivalence
of nonlinear Schro(cid:127)dinger equations and so on. Early in 1957, Suhl [131] had studied
the in(cid:12)nite dimensional dynamic system of the Landau{Lifshitz equations with
Gilbert damping term. A series of further studies on the theory of dynamics and
numerical results have appeared since then. In recent years, the ferromagnetic mate-
rials have been widely applied in the video and recording apparatus. This is one of
the applications for Landau{Lifshitz equations.
From 1982, mathematicians began their studies onthe well-posedness forLandau{
Lifshitz equations. In China, a group headed by Yulin Zhou and Boling Guo proved
the existence of the global weak solutions to the initial value problems and initial
boundary value problems for Landau{Lifshitz equations from one dimension to multi-
dimensions[150{157].AlougesandSoyeur[4]provedsimilarresultsbypenaltymethod
in1992. We alsoreferthe readers tothe result by PL.Sulem, C. Sulem andC. Bardos
[132]. Since then, many other results on the global existence were obtained [20{22].
However, the regularity and the uniqueness were unsolved in the 1980s due to the
complexity of Landau{Lifshitz equations.
However, in 1991, Zhou, Guo and Tan [158] obtained the existence and unique-
ness of global smooth solution to one-dimensional Landau{Lifshitz equations with
or without Gilbert damping by using a mobile frame on S2 and some (cid:12)ne a priori
estimates.
In 1993, Guo and Hong began the studies on two-dimensional Landau{Lifshitz
equations. They established in [77] the relations between two-dimensional Landau{
Lifshitz equations and harmonic maps and applied the approaches studying harmonic
maps to get the global existence and uniqueness of partially regular weak solution.
v
vi Landau-Lifshitz Equations
This conclusion has been cited by many others up to now and gives rise to many
successive works (see, for example, [47{49, 52, 89, 113, 115, 143]). Later on, in 1998,
Chen, Ding and Guo [29] further proved that all the weak solutions with (cid:12)nite energy
must be the Chen{Struwe solutions [34]. The uniqueness was also given. This says
that the weak solution with (cid:12)nite energy is globally smooth with exception of (cid:12)nitely
many singular points at most.
From1998to2001,GuoandDingdiscussed manyotherLandau{Lifshitzequations
such as inhomogeneous equations, unsaturated equations and compressible equations
[50, 51, 75, 108].
From the beginning of the new century, more and more mathematicians are
interested in the researches of Landau{Lifshitz equations. We refer the readers to
the works by Guo, Su, Carbou and Harpes, et al. [20{23, 80{84, 89] on Landau{
Lifshitz equations and Landau{Lifshitz{Maxwell equations.
A natural question is the regularity of weak solutions to the higher dimen-
sional Landau{Lifshitz equations. In this aspect, in 2004, Liu [109] proved that the
\stationary" weak solutions of higher dimensional Landau{Lifshitz equations are par-
tially regular. The Hausdor(cid:11) dimensions and the Hausdor(cid:11) measures of the singular
set were estimated. These extend the results on harmonic map heat (cid:13)ow by Feldman
[58] to Landau{Lifshitz equations. At the same time, Moser [115] obtained the similar
results for lower dimensional Landau{Lifshitz equations by di(cid:11)erent methods.
We know that the \stationary" conditions are hard to verify. So, in 2005, Melcher
[113] proved the partial regularity for the weak solutions to the initial value problems
of Landau{Lifshitz equations. However, as stated by Melcher, his method does not
(cid:12)t the other dimensional problems and, the partial regularity of weak solutions to the
boundary value problems are still unsolved.
This attracted the attention of Changyou Wang at the University of Kentucky.
Wang [143], using the method of [142], proved the partial regularity for the weak
solutions of the initial value problems and initial boundary value problems on three-
and four-dimensional manifolds.
However, all the results on the partial regularity only answered the questions on
the singular set such as how many points there are in the set or how large the set is,
provided thatthe singularity does exist. But, the existence of(cid:12)nite time singularity of
weak solutions is notanswered. Forthe harmonic mapheat (cid:13)ow, the similar questions
were answered by Chen and Ding [30] in 1990 (n (cid:21) 3) and by Chang, Ding and Ye
[26] (n = 2) in 1992 (see also [39] and many others).
Does the weak solution of Landau{Lifshitz equations really blow-up at (cid:12)nite time?
Pistella and Valente [123] in 2002, Bartels, Ko and Prohl [13] in 2005, gave positive
answers respectively by numerical analysis.
In 2007, Ding and Wang [52] rigorously proved that in three and four dimensions,
someDirichletproblemsandNeumannproblemsforLandau{Lifshitzequationsindeed
admit (cid:12)nite time blow-up solutions. Comparing with the similar proofs for harmonic
map heat (cid:13)ows, we do not have the monotonicity inequality and the Bochner identity.
Preface vii
Unfortunately, our method does not apply to two-dimensional problems and higher
dimensional problems. We do not know the types of the singularities either.
In recent years, there are many papers discussing the other problems such as
domainwall, energyconcentrations andvortices. Wereferto[44{46, 98{100,116,117]
and references therein.
The aim of this book is to introduce the readers the key works of the group headed
by Yulin Zhou and Boling Guoin China from 1980s. There is a bibliography comment
at the end of every chapter to introduce other mathematicians’ works in this (cid:12)eld and
brie(cid:13)y state the development in recent years. However, it is not possible to include all
the works and achievements throughout the world in such a short comment behind
every chapter.
The authors are deeply grateful to Professor Yulin Zhou for his support to this
work. We also want to thank Professors Yongqian Han, Jianqing Chen for their
help in the preparation of this book. This work was partially supported by the
projects of Natural Science Foundation of China (Grant No. 10471050), the National
973 Program of China (Grant No. 2006CB805902), Natural Science Foundation of
Guangdong Province (Grant No. 7005795) and University Special Research Fund for
Ph.D. Program of China (Grant No. 20060574002).
The Authors
March 2007
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
Preface v
1 Spin Waves and Equations of Ferromagnetic Spin Chain 1
1.1 Physics Background for the Equations of Ferromagnetic Spin Chain . 1
1.2 A Simple Derivation of Landau{Lifshitz Equation . . . . . . . . . . . 3
1.3 Equations for the Antiferromagnets . . . . . . . . . . . . . . . . . . . 10
1.4 Spin Waves in Ferromagnets . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Spin Waves in Antiferromagnets . . . . . . . . . . . . . . . . . . . . . 22
1.6 Bibliography Comments . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Integrability of Heisenberg Chain 35
2.1 Spin Waves and Solitary Waves . . . . . . . . . . . . . . . . . . . . . 35
2.2 Geometric Representation for the Landau{Lifshitz Equations . . . . . 40
2.3 Inhomogeneous Heisenberg Chain . . . . . . . . . . . . . . . . . . . . 43
2.4 Spherical (Cylindrical) Symmetric Heisenberg Equations of
Ferromagnetic Spin Chain . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5 Bibliography Comments . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 One-Dimensional Landau{Lifshitz Equations 53
3.1 Initial Boundary Value Problem of One-dimensional Ferromagnetic
Spin Chain Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Nonlinear Initial-boundary Value Problem for the System of
Ferromagnetic Spin Chain . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3 Smooth Solution for the Ferromagnetic Spin Chain Systems . . . . . 87
3.4 Smooth Solution for the 1D Inhomogeneous Heisenberg Chain
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.5 Measure-Valued Solution to the Strongly Degenerate Compressible
Heisenberg Chain Equations . . . . . . . . . . . . . . . . . . . . . . . 114
3.6 Bibliography Comments . . . . . . . . . . . . . . . . . . . . . . . . . 121
4 Landau{Lifshitz Equations and Harmonic Maps 123
4.1 Weak Solution to Multidimensional Ferromagnetic Spin Chain
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
ix